Xdc.sevsample function

Likelihood-Ratio-Test Statistics: Several Sample Dirichlet-Multinomial Test Comparison

This routine provides the value of the Likelihood-Ratio-Test Statistics and the corresponding p-value for evaluating the several sample Dirichlet-Multinomial parameter test comparison.

Xdc.sevsample(group.data, epsilon = 10^(-4), est = "mom")

Arguments

• group.data: A list where each element is a matrix of taxonomic counts(columns) for each sample(rows). (See Notes 1 and 2 in details)
• epsilon: Convergence tolerance. To terminate, the difference between two succeeding log-likelihoods must be smaller than epsilon. Default value is 10^(-4).
• est: The type of parameter estimator to be used with the Likelihood-ratio-test statistics, 'mle' or 'mom'. Default value is 'mom'. (See Note 3 in details)

Returns

A list containing the Xdc statistics and p-value.

Details

To assess whether the Dirichlet parameter vector, $\mathbf{\alpha}_{\mathrm{m}}=\mathbf{\pi}_{\mathrm{m}} \frac{1-\theta_{\mathrm{m}}}{\theta_{\mathrm{m}}}$(a function of the RAD probability-mean vector and overdispersion), observed in $J$ groups of microbiome samples are equal to each other, the following hypothesis $\mathrm{H}_{\mathrm{o}}: \mathbf{\alpha}_{\mathrm{1}} = \cdots =\mathbf{\alpha}_{\mathrm{m}}=\cdots= \mathbf{\alpha}_{\mathrm{J}}=\mathbf{\alpha}_{\mathrm{o}}$

versus $\mathrm{H}_{\mathrm{a}}: \mathbf{\alpha}_{\mathrm{m}} \ne \mathbf{\alpha}_{\mathrm{o}}, m=1, \ldots, J$ can be tested. The null hypothesis implies that the HMP samples across groups have the same mean and overdispersion, indicating that the RAD models are identical. In particular, the likelihood-ratio test statistic is used, which is given by,

The asymptotic null distribution of $x_{\mathrm{dc}}$ follows a Chi-square with degrees of freedom equal to (J-1)*K, where K is the number of taxa (Wilks, 1938).

1. Note 1: The matrices in group.data must contain the same taxa, in the same order.
2. Note 2: Each taxa should be present in at least 1 sample, a column with all 0's may result in errors and/or invalid results.
3. Note 3: 'mle' will take significantly longer time and may not be optimal for small sample sizes; 'mom' will provide more conservative results in such a case.

References

Wilks, S. S. (1938). The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses. The Annals of Mathematical Statistics 9, 60-62.

Examples

data(saliva) 
        data(throat)
        
        ### Combine the data sets into a single list
        group.data <- list(saliva, throat)
        
        xdc <- Xdc.sevsample(group.data)
        xdc